TITLE: Cusp forms with integer Fourier-coefficients
QUESTION [9 upvotes]: Let $S_k(\Gamma_1(N))(\mathbb{Z})$ be the set of modular forms of weight $k$ and level $N$ with integer Fourier coefficients. Then is true that any cusp form can be written as $\mathbb{Q}$ linear combination of Hecke eigenforms with integer coefficients ?

REPLY [10 votes]: No. Take $k = 24$ and $N = 1$. Then $\Delta^{2} = q^{2} - 48q^{3} + 1080q^{4} + \cdots \in S_{K}(\Gamma_{1}(N),\mathbb{Z})$. However, if we write $\Delta^{2} = c_{1} f_{1} + c_{2} f_{2}$, where $f_{1}$ and $f_{2}$ are the Hecke eigenforms (with coefficients in $\mathbb{Q}(\sqrt{144169})$) then (if we order $f_{1}$ and $f_{2}$ appropriately), $c_{1} = \frac{1}{24 \sqrt{144169}}$ and $c_{2} = -\frac{1}{24 \sqrt{144169}}$.